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In recent years, Lengyel, Komatsu and Young, Leonetti and Sanna, and Adelberg made some progress on the $p$-adic valuations of $s(n,k)$. In this paper, by introducing the concept of $m$-th Stirling numbers of the first kind and providing a detailed 2-adic analysis, we show an explicit formula on the 2-adic valuation of $s(2^n, k)$. We also prove that $v_2(s(2^n+1,k+1))=v_"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1812.04539","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-12-11T16:56:58Z","cross_cats_sorted":[],"title_canon_sha256":"817f198e35e584b7025f785bd89ef687e91b42fd3b518b2a9b3539dc21ca4fdb","abstract_canon_sha256":"dce17cbd979e17a6d933eb63a337933bc292b359a2a5235ac840b03d554b3c5e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:58:31.772708Z","signature_b64":"eNTtXWNP8DNCkizh+EMSdGHNnYt9JZ9yFPuGEX29YPPqUTZUa9gZTuAmrc0gnmiolx+FXUE4oEnsQwfvlADGCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"64ad510e7d90aa7a14df1e542a218a7e1fbe17f5f50068619a48bffe170e85b7","last_reissued_at":"2026-05-17T23:58:31.772094Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:58:31.772094Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The 2-adic valuations of Stirling numbers of the first kind","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Min Qiu, Shaofang Hong","submitted_at":"2018-12-11T16:56:58Z","abstract_excerpt":"Let $n$ and $k$ be positive integers. We denote by $v_2(n)$ the 2-adic valuation of $n$. The Stirling numbers of the first kind, denoted by $s(n,k)$, counts the number of permutations of $n$ elements with $k$ disjoint cycles. In recent years, Lengyel, Komatsu and Young, Leonetti and Sanna, and Adelberg made some progress on the $p$-adic valuations of $s(n,k)$. In this paper, by introducing the concept of $m$-th Stirling numbers of the first kind and providing a detailed 2-adic analysis, we show an explicit formula on the 2-adic valuation of $s(2^n, k)$. We also prove that $v_2(s(2^n+1,k+1))=v_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.04539","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1812.04539","created_at":"2026-05-17T23:58:31.772178+00:00"},{"alias_kind":"arxiv_version","alias_value":"1812.04539v1","created_at":"2026-05-17T23:58:31.772178+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.04539","created_at":"2026-05-17T23:58:31.772178+00:00"},{"alias_kind":"pith_short_12","alias_value":"MSWVCDT5SCVH","created_at":"2026-05-18T12:32:40.477152+00:00"},{"alias_kind":"pith_short_16","alias_value":"MSWVCDT5SCVHUFG7","created_at":"2026-05-18T12:32:40.477152+00:00"},{"alias_kind":"pith_short_8","alias_value":"MSWVCDT5","created_at":"2026-05-18T12:32:40.477152+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MSWVCDT5SCVHUFG7DZKCUIMKPY","json":"https://pith.science/pith/MSWVCDT5SCVHUFG7DZKCUIMKPY.json","graph_json":"https://pith.science/api/pith-number/MSWVCDT5SCVHUFG7DZKCUIMKPY/graph.json","events_json":"https://pith.science/api/pith-number/MSWVCDT5SCVHUFG7DZKCUIMKPY/events.json","paper":"https://pith.science/paper/MSWVCDT5"},"agent_actions":{"view_html":"https://pith.science/pith/MSWVCDT5SCVHUFG7DZKCUIMKPY","download_json":"https://pith.science/pith/MSWVCDT5SCVHUFG7DZKCUIMKPY.json","view_paper":"https://pith.science/paper/MSWVCDT5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1812.04539&json=true","fetch_graph":"https://pith.science/api/pith-number/MSWVCDT5SCVHUFG7DZKCUIMKPY/graph.json","fetch_events":"https://pith.science/api/pith-number/MSWVCDT5SCVHUFG7DZKCUIMKPY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MSWVCDT5SCVHUFG7DZKCUIMKPY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MSWVCDT5SCVHUFG7DZKCUIMKPY/action/storage_attestation","attest_author":"https://pith.science/pith/MSWVCDT5SCVHUFG7DZKCUIMKPY/action/author_attestation","sign_citation":"https://pith.science/pith/MSWVCDT5SCVHUFG7DZKCUIMKPY/action/citation_signature","submit_replication":"https://pith.science/pith/MSWVCDT5SCVHUFG7DZKCUIMKPY/action/replication_record"}},"created_at":"2026-05-17T23:58:31.772178+00:00","updated_at":"2026-05-17T23:58:31.772178+00:00"}